Optimal regularity for phase transition problems with convection

Abstract

In this paper we consider a steady state phase transition problem with given convection $\mathbf{v}$. We prove, among other things, that the weak solution is locally Lipschitz continuous provided that $\mathbf{v}=D\xi$ and $\xi$ is a harmonic function. Moreover, for continuous casting problem, i.e. when $\mathbf{v}$ is constant vector, we show that Lipschitz free boundaries are $C^1$ regular surfaces.